Global Dynamics of Rational Difference Equations and

Abstract

Global dynamics of a system of nonlinear difference equations was investigated, which had five kinds of equilibria including isolated points and a continuum of nonhyperbolic equilibria along the coordinate axes. The local stability of these equilibria was analyzed which led to nine regions in the parameters space. The solution of the system converged to the equilibria or the boundary point or in each region depending on nonnegative initial conditions. These results completely described the behavior of the system.

1. Introduction

In this paper, we focus on the global dynamics of the following system:where the parameters and are positive and the initial conditions and are nonnegative.

In [1], the stability of (1) was investigated. If and , the equilibrium of (1) is globally asymptotically stable. If and , the equilibria and of (1) are locally unstable. The global dynamics of (1) was considered only for the case and .

System (1) can be regarded as a generalization of the equationwith the parameter being positive and initial conditions ,   being nonnegative, which was studied in [2] on the stability of the equilibria, nonexistence of prime period-two solutions, and global dynamics of the equation. More accurate results were obtained in our forthcoming work that every positive solution of (2) converged to its equilibria, for and for .

The above equations and systems are also the special cases of a general equationwith nonnegative parameters and proper initial conditions. Several global asymptotic results for some special cases of (3) were obtained in [3–13].

As for the definition of stability and the method of linearized stability, see [1–21]. For other types of equations and systems, see [14–19, 22–39]. As for the definition of basin of attraction and the stable manifold and so on, see [35–39].

In this article, we try to determine a complete picture of the behavior of (1). First, we part completely the regions of parameters by equilibria. Second, by the theory of linearized stability, we describe the local stability of these equilibria for five cases and derive nine regions in plane. At last, we present the main results on global dynamics of (1) in these regions.

It is the first time that the parameters spaces are divided into nine regions and complex dynamics of (1) are derived according to the initial conditions for each region. It is also the first time that we determine the details that the equilibrium is nonhyperbolic or a saddle point if it is unstable. It is worth pointing out that the system has a continuum of nonhyperbolic equilibria along a vertical line or/and a horizontal line, which lead to interesting phenomena on the global dynamics.

2. Existence of Equilibria

The study of dynamics of difference equations often requires that equilibria be calculated first, followed by a local stability analysis of the equilibria. This is then complemented by other considerations (existence of periodic points, etc.). If the analysis is applied to a class of equations dependent on one or more parameters, the task is complicated by the fact that a formula is not always available for equilibria, and even if it is, determination of stability of parameter-dependent equilibria may be a daunting task.

First of all, we obtain the existence of equilibria of (1).

As is known, the equilibrium of (1) satisfies from which it follows that one of five cases holds for the equilibrium points of (1):(1) if one of the following conditions holds:(i) and .(ii) and .(iii) and .(2) and if and .(3) and with arbitrary if and .(4) and with arbitrary if and .(5) and and with arbitrary and if and .

3. Local Stability of Equilibria

Now, we consider the local stability of these equilibria of (1).

3.1. Local Stability of

Theorem 1. Suppose that is the equilibrium of (1). Then one of the following holds: (1) is locally asymptotically stable for and .(2) is nonhyperbolic of the stable type for one of the following three cases: Case  2.1: ,  . Case  2.2: ,  . Case  2.3: ,  .(3) is nonhyperbolic of the unstable type for one of the following two cases: Case  3.1: ,  . Case  3.2: ,  .(4) is a saddle point for one of the following three cases: Case  4.1: and . Case  4.2: and . Case  4.3: and .

Proof. The linearized system of (1) about isAs is shown in [1], we have its characteristic polynomialHere, we only focus on one of these two factors. Obviously, we have Thus, the distribution of solutions of is one of the following: (i)Two real roots in for (ii)One root being and the other being for (iii)One root in and the other in for By Theorem  1.2.1 in [21], we obtain the conclusions and complete the proof.

3.2. Local Stability of

Now, we consider the local stability of the positive equilibrium of (1), which exists only for and .

Theorem 2. Assume that and and is the positive equilibrium of (1); then is a saddle point.

Proof. The linearized equation of (1) about iswhere . It is obvious that for and .
As is shown in [1], its characteristic polynomial isWe have , , , and for . Thus, has one solution in and one in .
Now, we divide it into two cases to show the distribution of the other two solutions of .
Case  1 (). In view of , we conclude that has three solutions in and one in and thus is a saddle point by Theorem  1.2.1 in [21].
Case  2 (). In this case, we rewrite (9) as follows:where , , and satisfyingNext, we try to prove all roots of the equation to be inside the unit disk for , which is necessary and sufficient to prove by Theorem  1.2.2 in [21].
First, we try to show for .
From (14), it is obvious and thus is equivalent to . In view of for and , we try to prove . To this end, we try to determine the exact range of .
In view of , from and , we could obtain . Thus, we have and is proved.
Second, we show for . To this end, we only need to show for .
From (11), we obtain and thus as desired.
In fact, more precisely, in view of , we have that for and for .
Therefore, for , we have ; for and for . Hence, we obtain for and thus all roots of the equation lie inside the unit disk. By Theorem  1.2.1 in [21], of (1) is a saddle point for .
Thus, we conclude that of (1) is a saddle point if it exists and we complete the proof.

3.3. Local Stability of

Now, we consider the local stability of the equilibria    of (1), which exists only for and .

Theorem 3. Assume that and and    are the equilibria of (1).
If or and , then is nonhyperbolic of the stable type.
If and , then is nonhyperbolic of the unstable type.

Proof. The linearized equation of (1) about every isfrom which we have its characteristic polynomialIt is obvious that has two solutions and .
Similarly, if and , then has two solutions and .
If or and , then has two solutions in .
If and , then has one solution in and the other in .
By Theorem  1.2.1 in [21], we derive the conclusions and complete the proof.

3.4. Local Stability of

Similar to the proof of Theorem 3, we have the following theorem.

Theorem 4. Assume that and and    are the equilibria of (1).
If or and , then is nonhyperbolic of the stable type.
If and , then is nonhyperbolic of the unstable type.

3.5. Local Stability of and

In case of , the equilibria of (1) include , , and   .

By Theorem 1, for , of (1) is nonhyperbolic of the stable type.

Similar to the proof of Theorem 3, the linearized equation of (1) about is (15) with and its characteristic polynomial is (16) with which has four roots in . It implies that every is nonhyperbolic of the stable type.

Similarly, every is nonhyperbolic of the stable type.

Theorem 5. Assume that , , , and   ( and ) are the equilibria of (1); then they are nonhyperbolic of the stable type.

There are 9 cases in parametric space with distinct local stability of distinct equilibria. We list the above results in Table 1. For simplicity, if of (1) is locally asymptotically stable, we denote If is nonhyperbolic, we denote If is nonhyperbolic of the stable type or the unstable type, we denote or N.H.. If is a saddle point, we denote .

Table 1 

Local stability of equilibria of (1).

4. Global Dynamics

For nonnegative initial conditions   , we assume that is the corresponding solution of (1). For simplicity, we often need to consider the behavior of and , respectively.

In the following, we try to investigate the global dynamics of (1) for these nine cases.

Case 1 (). By Theorem  1 in [1], of (1) is globally asymptotically stable for and ; that is, basin of attraction of of (1) is .

Case 2 (). In this case, of (1) is a saddle point for and .
If the initial conditions    are on -axis, we have for all , and system (1) is changed into a single equationfrom which we have with and satisfying the characteristic equation . For , one of the modulus of and is smaller than one and the other is greater than one. Therefore, we have with on -axis for .
If are on -axis, we have for all and thus for .
We declare that the stable manifold of is .
In fact, if in the first quadrant, then from (1), we obtainWe ascertain for . In fact, we could deduce that by comparison and the theory of linear difference equations.
Setting ,   andwe obtain for all by induction.
From (19), we have    with and satisfying the characteristic equation from which we have and for and thus goes to zero as tends to . Therefore, we have for .
Next, we consider the behavior of the component .
From the fact of , there is a positive constant satisfying such that for with being some positive integer. From (1), for , we obtainBy comparison and the theory of linear difference equations, we get for .
Hence, we obtain the following theorem.

Theorem 6. If and , then the global stable manifold of (1) is Whenever , then

Case 3 (). In this case, both and of (1) are saddle points for and .
We claim that sets of the formare invariant for sufficiently small : that is, for all if   .
Suppose and , from (1); then we havefrom which we have . By induction, we have for all andfrom which it follows that .
Similarly, sets of the formare invariant for sufficiently small . For , then we have .

Theorem 7. If and , then sets of the form and (defined by (21) and (24)) are invariant of (1) for sufficiently small .
If   , then .
If   , then .

Case 4 (). In this case, of (1) is a saddle point for and .
Similar to that of Case 2, we obtain the following theorem.

Theorem 8. If and , then the stable manifold of of (1) is Whenever , then

Case 5 (). In this case, and of (1) are nonhyperbolic of the stable type for and .
For on -axis, we have that for all and satisfies (17) with , from which it follows that exists depending on   .
For on -axis, we have that for all and satisfiesfrom which we have for .
For positive initial conditions, similar to Case 2, we also know for . Specially, it follows that for all for some positive integer . From (1), for , we havefor and thus exists. Thus, basin of attraction of is . If    then and exists.

Theorem 9. If and , then basin of attraction of of (1) is Whenever   , then

Case 6 (). In this case, and of (1) are nonhyperbolic for and .
For on -axis, we have that for all and satisfies (17) with , from which it follows that .
For on -axis, we have that for all and satisfies (25) with , from which it follows that exists depending on .
There is a curve such that the first quadrant is divided into two connected parts andIf , then we have for all . Thus, from (1), we obtainBy comparison and the results of (2), we have for .
Hence, similar to Case 2, we have for .
If , then we also obtain the above conclusion.
If , that is,   , then we choose such a that : that is, . There is a curve ,which is below the curve such that is divided into two connected partsIf , that is,   , then we have for all by induction. Thus, from (1), we obtainfor arbitrarily small . Thus, we conclude that . Specially, we have for for some positive integer . Hence, from (1) we also obtain (25) with for , from which it follows that exists depending on .
If on the curve , then we also derive the above conclusion.
Thus, we obtain the following theorem.

Theorem 10. If and , then of (1) is nonhyperbolic of the unstable type and every    of (1) is nonhyperbolic. More precisely, is nonhyperbolic of the unstable type for and is nonhyperbolic of the stable type for .
There is a curve defined by (27) such that for on and above the curve .
There is a curve defined by (31) with such that with for on and below the curve .

Case 7 (). In this case, and of (1) are nonhyperbolic for and .
Similar to that of Case 6, we obtain the following theorem.

Theorem 11. If and , then of (1) is nonhyperbolic of the unstable type and every    of (1) is nonhyperbolic. More precisely, is nonhyperbolic of the unstable type for and is nonhyperbolic of the stable type for .
There is a curve defined by (27) such that for    on and below the curve .
There is a curve defined by such that with for    on and above the curve with .

Case 8 (). In this case, both and of (1) are nonhyperbolic of the stable type for and .
Similar to that of Case 5, we obtain the following theorem.

Theorem 12. If and , then basin of attraction of of (1) is Whenever   , then .

Case 9 (). Here, , , and   ( and ) of (1) are nonhyperbolic of the stable type for .
Now, we focus on . There is a curve (defined by (27)) passing through such that the first quadrant is divided into two connected parts andIf   () are on the curve , system (1) is reduced to a single equation and every positive solution of (1) converges to .
If , we have   . By induction, from (1), we have for all . For , from (1), we haveand thus by comparison and the results of (2). Specially, it follows that for all for some positive integer . Thus, from (1), we have (25) for and hence exists depending on initial conditions. Therefore, we have    for .
Similarly, we obtain    for .

Theorem 13. If and , then basin of attraction of of (1) is If   , then .
If   , then .
Here, and are defined by (34) and (35), respectively.

These above theorems completely describe the global dynamics of (1) and are listed in Table 2. The solution of system (1) converged to either the equilibria or the boundary point or depending on nonnegative initial conditions and parameters.

Table 2 

Global dynamics of (1).

5. Conclusion

It is known that the techniques in the investigation of the behavior of difference equations can be used in investigating equations arising in mathematical models describing real life situations in biology, economics, physics, sociology, control theory, and vice versa.

In this paper, we investigate a system of nonlinear difference equations, which has complex dynamics. We use the results of linearized stability to analyze the local stability of five kinds of equilibria of (1) with parameters in nine regions in the first quadrant. Specially, for particular parameters and , dynamics of the system is very interesting with a continuum of nonhyperbolic equilibria along a line. Generally speaking, the solution of (1) converges to its equilibria if the equilibrium is locally asymptotically stable or nonhyperbolic of the stable type, depending on initial conditions. Otherwise, it may converge to the boundary point or or exhibit somewhat chaos.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.